622 Worked-Out Solutions to Exercises: Chapters 11 to 19
- We must simplify the inequality so that y appears all by itself on the left side of the
“not equal” symbol, and a plain numeral appears all by itself on the right. Here’s the
inequality again, for reference:
y/2≠ 4 y+ 7
First, let’s multiply through by 2. That gives us
(y/2)× 2 ≠ (4y+ 7) × 2
which multiplies out to
y≠ 8 y+ 14
Now, let’s subtract 8y from each side. That gives us
y− 8 y≠ 8 y+ 14 − 8 y
which simplifies to
− 7 y≠ 14
We can divide this through by −7 to get
(− 7 y)/(−7)≠ 14 /(−7)
which simplifies to
y≠− 2
The original inequality holds true for all values of y except −2.
- We must simplify the inequality so that z appears all by itself on the left side of the
“smaller than or equal” symbol, and a plain numeral appears all by itself on the right.
Here’s the inequality again, for reference:
z/(−3)≤ 6 z+ 6
Let’s multiply through by −3, remembering that we must reverse the sense of the inequal-
ity whenever we multiply through by a negative. That gives us
[z/(−3)]× (−3)≥ (6z+ 6) × (−3)
which simplifies to
z≥− 18 z− 18